1
$\begingroup$

Let $S=\langle n_1,...,n_r \rangle$ be a commutative semigroup, and let $I_S \subset k[x_1,...,x_r]$ the associated ideal of $S$, defined as the kernel of the polinomial map $\varphi:k[x_1,...x_n] \rightarrow k[t]$, defined by $\varphi(x_i)=t^{n_i}$ for all $i = 1,2,...,r$. It is known that for $r=3$ the ideal $I_S$ is a complete intersection or is generated by the 2x2 minors of a 2x3 matrix like $$ \begin{pmatrix} {x_1}^{a} & {x_2}^{b} & {x_3}^{c}\\ {x_2}^{d} & {x_3}^{e} & {x_1}^{f}\\ \end{pmatrix} $$ and so $I_S$, if is not a complete intersection, is a determinantal ideal. So I would ask:

  1. There are some conditions to establish when $I_S$ is a determinantal ideal for $r\geq4$ ?
  2. Is it possible generalize the case $r=3$? And so it could appens that for $r\geq4$, $I_S$ is the ideal generated by the 2x2 minors of a (r-1) x r matrix (or in general a (r-1) x m matrix)?
$\endgroup$
  • $\begingroup$ For a general $(r-1)\times r$ matrix $A$ with entries in $k[x_{1},\ldots ,x_{r}]$, the locus where $\operatorname{rk}A\leq 1 $ has codimension $(r-1)(r-2)$ in $\mathbb{A}^r$. Therefore it is indeed a curve for $r=3$, but it is empty for $r\geq 4$. So the answer to 2. is very probably no — unless you find a very particular matrix $A$. $\endgroup$ – abx Jan 31 at 15:32

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.